Knowledge on a Budget

TL;DR

This paper introduces semiring-annotated topological spaces to model resource-aware evidence acquisition in epistemic logic, enabling reasoning about what is observable within resource constraints.

Contribution

It extends Topological Evidence Logic by incorporating resource annotations, providing a formal framework for resource-bounded epistemic reasoning.

Findings

Developed a family of seat-based epistemic logics with resource-indexed modalities.

Provided sound and strongly complete axiomatisations for the new logics.

Introduced notions of bisimulation and disjoint union to analyze expressive power.

Abstract

In various computational systems, accessing information incurs time, memory or energy costs. However, standard epistemic logics usually model the acquisition of evidence as a cost-free process, which restricts their applicability in environments with limited resources. In this paper, we bridge the gap between qualitative epistemic reasoning and quantitative resource constraints by introducing semiring-annotated topological spaces (seats). Building on Topological Evidence Logic (TEL), we extend the representation of evidence as open sets, adding an annotation function that maps evidence to semiring ideals, representing the resource budgets sufficient for observation. This framework allows us to reason not only about what is observable in principle, but also about what is affordable given a specific budget. We develop a family of seat-based epistemic logics with resource-indexed modalities and provide sound, strongly complete axiomatisations for these logics. Furthermore, we introduce suitable notions of bisimulation and disjoint union to delineate the expressive power of our framework.